DOI: https://doi.org/10.15588/1607-3274-2017-1-11

### DECOMPOSITION OF SUM OF GAUSSIANS DISTORTED BY IMPULSIVE NOISE

#### Abstract

noise, leads to considerable errors. The object of this study is the process of Gaussian decomposition in an impulsive noise environment.

Objective. The goal of this work is the development of a method of Gaussian decomposition for the case when the data are distorted by

impulsive noise.

Method. The proposed method of Gaussian decomposition is based on solving the problem of unconstrained minimization the objective

function by unknown parameters. The problem statement is built on the criterion of a minimum extent which is used to the solution residual.

Process of Gaussian decomposition is implemented iteratively by successive selecting from the sum of Gaussian functions such a Gaussian function, which initially is the most extended, and then the one which is less extended, etc. To determine the parameter values of Gaussian function the two approaches are described. The first approach is based on the iterative method, which is used for solving the set of nonlinear equations, derived from the necessary conditions for the minimum of objective function. The second approach is based on the method of

passive searching of objective function minimum, where the test points are chosen from the condition that in these points the discrepancy is

zero. It is indicated that the second approach has a wider range of applicability than the first one. On the basis of the second approach, an

iterative algorithm is built. The way of selecting the initial parameter values of Gaussian function under impulsive noise environment is

presented. Also, the rule for choosing the best values of Gaussian parameters and the rule to stop of computing are formulated.

Results. Simulations for the problems of the single Gaussian curve fitting to data and of the Gaussian decomposition for the sum of five

Gaussians in the case, when the data are distorted by Cauchy noise, confirmed the performance of the proposed method.

Conclusions. The proposed method is the efficient tool of Gaussian decomposition for the sum of Gaussians distorted by the impulsive

noise with Cauchy distribution. It can be expanded to the case when the functions have other shape and other unknown parameters.

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